Question: "Can someone please explain to me why the..."

fellesturduj

Answered question

2022-03-19

Can someone please explain to me why the following identity is true?

lim_{x \to \infty} {(1 + \frac{a}{x})}^{x} = e^{a}

obduciramwz6

Beginner2022-03-20Added 8 answers

You have to recall the fundamental limit

lim_{x \to \pm \infty} {(1 + \frac{1}{x})}^{x} = e

Think of it as a general rule like this:

lim_{⋆ \to \pm \infty} {(1 + \frac{1}{⋆})}^{⋆} = e

where the star can be substituded by any expression (which tends to

\pm \infty

).
So

lim_{x \to \pm \infty} {(1 + \frac{a}{x})}^{x} = lim_{x \to \pm \infty} {[{(1 + \frac{1}{\frac{x}{a}})}^{\frac{x}{a}}]}^{\frac{a}{x} \cdot x} = e^{a}

Jeffrey Jordon

Expert2022-07-05Added 2605 answers

Answer is given below (on video)

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